# Video: AQA GCSE Mathematics Higher Tier Pack 5 • Paper 1 • Question 18

This prism has a cross section consisting of a semicircle attached to a parallelogram. All the lengths in the diagram are in centimetres. Find an expression for the total volume of the prism in cubic centimetres. Write your answer in the form 𝑎(𝑥³ + 𝑥²) + 𝑏𝜋𝑥³.

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### Video Transcript

This prism has a cross section consisting of a semicircle attached to a parallelogram. All the lengths in the diagram are in centimetres. Find an expression for the total volume of the prism in cubic centimetres. Write your answer in the form 𝑎 multiplied by 𝑥 cubed plus 𝑥 squared plus 𝑏 multiplied by 𝜋 multiplied by 𝑥 cubed.

Let’s begin by reminding ourselves what the formula is for the volume of a prism. It’s the area of its cross section multiplied by its length. The cross section of this prism is a compound shape. It’s made up of a semicircle and a parallelogram. We do know its length though. It’s one-half 𝑥.

Let’s begin by working out the area of the cross section. We’re going to calculate the area of the two shapes it’s made up of. The formula for area of a circle is 𝜋𝑟 squared, where 𝑟 is the radius. So we can find the area of a semicircle by halving this. We can find the diameter of our semicircle. Since opposite sides of a parallelogram are equal in length, this means the diameter of our semicircle is 𝑥. The radius is half of the length of the diameter. So it’s a half of 𝑥 or 𝑥 over two. The area of the semicircle is therefore given by 𝜋 multiplied by the radius squared. That’s 𝑥 over two squared all over two.

We need to be careful when squaring 𝑥 over two. We have to square both the numerator and the denominator. That gives us 𝑥 squared over four. Having a fraction within a fraction made that last sum a little bit tricky. Instead, if we write it as a half of 𝜋 multiplied by 𝑥 squared over four, it’s much easier to deal with.

We multiply the numerators. That’s one multiplied by 𝜋 multiplied by 𝑥 squared. That’s 𝜋𝑥 squared. We then multiply the denominators, remembering that the denominator of 𝜋 will be one. Two multiplied by one multiplied by four is eight. So the area of the semicircle is 𝜋 multiplied by 𝑥 squared over eight or an eighth of 𝜋𝑥 squared.

Next, we need to find the area of the parallelogram. That’s found by multiplying its base by its perpendicular height. The base of our parallelogram is 𝑥 and its height is 𝑥 plus one. So its area is 𝑥 multiplied by 𝑥 plus one. It’s important to include the brackets around the 𝑥 plus one because that reminds us that the 𝑥 is multiplying both the 𝑥 on the inside of that bracket and by the one. 𝑥 multiplied by 𝑥 is 𝑥 squared and 𝑥 multiplied by one is 𝑥. So the area of the parallelogram is 𝑥 squared plus 𝑥.

We can see then that the total area of the cross section is an eighth of 𝜋 multiplied by 𝑥 squared plus another 𝑥 squared plus 𝑥. All that’s left is to multiply this area by the length of the prism. That’s one-half 𝑥. We’ll multiply this bracket out by multiplying each term inside the bracket by one-half 𝑥. One-eighth multiplied by one-half is one sixteenth. So we get one sixteenth of 𝜋𝑥 cubed. 𝑥 squared multiplied by a half 𝑥 is a half 𝑥 cubed and 𝑥 multiplied by a half 𝑥 is a half 𝑥 squared.

We do need to write our answer in the form given. Notice how 𝑥 cubed and 𝑥 squared both have a coefficient of a one-half. We can therefore factorise this partially by taking out a factor of one-half. And a half 𝑥 cubed plus a half 𝑥 squared is the same as a half multiplied by 𝑥 cubed plus 𝑥 squared. And we’re left also with one sixteenth of 𝜋𝑥 cubed. Comparing this to our form given in the answer, we can see that 𝑎 is equivalent to a half and 𝑏 is equivalent to one sixteenth.

And we’re done! The volume of our prism is a half multiplied by 𝑥 cubed plus 𝑥 squared plus a sixteenth 𝜋𝑥 cubed.